218 
MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 
Substituting these values in the right-hand member of (12), we obtain 
0/x 1_ — 4:Co-fjr 0/i X" — 4(y"yLt^ d(f) J 
2cofjiIi 
X" — 4co^/j,“ d/u. 
i\h 
(1 - (X2 - 4a,V) j 
_ 
2„2\ f 
( 16 ). 
Hence, provided that X be not equal to zero, we have 
_ 0 [ h(l — /u,-) 0-vp 2fo/iX 
~ 011.1 >.2 _ 4.,>3,,2 017 ■" oX r>i.3 - 4r„2,,2 
01^ 
0yLi|X“ — 4(o~/j.~ 0/A aX (X“ — 4o)”iJ?) d(p 
0 f 2a)/iX 
+ 
dir 
A 
^'l 
0^ [ aX (X^ — 4a,~/A^) 0/6 (1 — /A®) (X“ — 4aj“/A^) 0^ J 
■ • (17)- 
This equation, in conjunction with the pressure equation (9) of § 2, serves to determine 
i//, ^ in terms of /a, (j). It is equivalent to the well-known equation used by Laplace 
in the ‘ Mecanique Celeste.’'"' Omitting from consideration for the present the types 
of motion defined by X = 0, we propose in the present paper to discuss only those 
solutions which are symmetrical with respect to the axis of rotation. In order that 
such solutions may exist, we must suppose that h, i// are independent of the 
equation (17) will then reduce to 
( 18 ), 
where for brevity we have put X/2co = f. 
0 
0/6 
A (1 — /A“) 0-Vp 
r - Sa 
I = 4aV^{ 
§ 5. Integ7'ation hy Means of Zonal Harmonics. 
Suppose that ^ is expressible as a series of zonal harmonics of the form 
C = X C„P„ (/a). 
U=1 
Neglecting the ellipticity of the surface, we may at once write down the value at 
the surface of the potential due to this distribution ; we have, namely, 
® ='‘y4Tl:C.p.(/.), 
)i=l 4 - A 
* Part I., Book IV., § o. 
